On Analytic Families of Invariant Tori for Pdes

Abstract. — We propose to apply a version of the classical Stokes expansion method to the perturbative construction of invariant tori for PDEs corresponding to solutions quasiperiodic in space and time variables. We argue that, for integrable PDEs all but finite number of the small divisors arising in the perturbative analysis cancel. As an illustrative example we establish such cancellations for the case of KP equation. It is proved that, under mild assumptions about decay of the magnitude of the Fourier modes all analytic families of finite-dimensional invariant tori for KP are given by the Krichever construction in terms of thetafunctions of Riemann surfaces. We also present an explicit construction of infinite dimensional real theta-functions and corresponding quasiperiodic solutions to KP as sums of infinite number of interacting plane waves.